On the Stability of the Nonlinear Center under Quasi-periodic Perturbations

Abstract

The problem of stability of the zero solution of a system with a “center”-type critical point at the origin of coordinates is considered. For the first time, such a problem for autonomous systems was investigated by A.M. Lyapunov. We continued Lyapunov’s investigations for systems with a periodic dependence on time. In this paper, systems with a quasi-periodic time dependence are considered. It is assumed that the basic frequencies of quasi-periodic functions satisfy the standard Diophantine-type condition. The problem under consideration can be interpreted as the problem of stability of the state of equilibrium of the oscillator $$\ddot {x} + {{x}^{{2n - 1}}}$$ = 0, n is an integer number, n ≥ 2, under “small” quasi-periodic perturbations.